Dynamics of Partial Differential Equations

Volume 9 (2012)

Number 2

A class of Neumann type systems and its application

Pages: 147 – 171

DOI: http://dx.doi.org/10.4310/DPDE.2012.v9.n2.a4

Author

Jinbing Chen (Department of Mathematics, Southeast University, Nanjing, Jiangsu, China)

Abstract

A class of Neumann type systems are derived separating the spatial and temporal variables for the 2+1 dimensional Caudrey-Dodd-Gibbon- Kotera-Sawada (CDGKS) equation and the modified Korteweg-de Vries (mKdV) hierarchy. The Lax-Moser matrix of Neumann type systems is worked out, which generates a sequence of integrals of motion and a hyperelliptic curve of KdV type. We deduce the constrained Hamiltonians to put Neumann type systems into canonical Hamiltonian equations and further complete the Liouville integrability for the Neumann type systems. We also specify the relationship between Neumann type systems and infinite dimensional integrable systems (IDISs), where the involutivity solutions of Neumann type systems yield the finite parametric solutions of IDISs. From the Abel-Jacobi variables, the evolution behavior of Neumann type flows are shown on the Jacobian of a Riemann surface. Finally, the Neumann type flows are applied to produce some explicit solutions expressed by Riemann theta functions for the 2+1 dimensional CDGKS equation and the mKdV hierarchy.

Keywords

Neumann type system, Dirac-Poisson bracket, Liouville integrability, Abel-Jacobi variable, explicit solution

2010 Mathematics Subject Classification

35Q51, 37K10, 37K20

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